Imagine that you've just moved into a large apartment building. Its rules specifically prohibit pets of any kind. Perhaps that's even part of why you chose the place, owing to allergies or an aversion to animals. Then, when your neighbor opens a door, you happen to see cats! What can you do?
Many co-ops and condos large and small cities explicitly ban pets (with the mandatory exception of service dogs). Some ban only specific kinds of pets, like cats and dogs, but allow less disruptive pets like fish. Many rental agreements and sublets can also prevent a tenant from having a pet in the particular apartment being rented, even if the building itself allows pets.
First, double-check the rules set forth in your lease or rental agreement to make sure that you're correct about the building prohibiting cats or other pets. If a neighbor is indeed violating the terms of the building rules, you have essentially two choices: you can confront the person, or you can go straight to the building management.
While you normally might seek to negotiate with a neighbor before bringing in a third party, there is little to negotiate in a situation like this. The neighbor wants to keep the pets, you most likely want them removed, especially given any allergies.
People are often defensive about their pets. Asking someone to give away cats will surely result in a hostile relationship, even if the person is in violation of the building rules. Keep in mind that you'll see your neighbor constantly: in the laundry room, in the elevator, and in the local grocery store. You might do best to try to avoid this awkwardness.
If you bring your concern to the building management, either in a letter or by calling its central office, the management might be able to intercede without using your name. In your communications, you should emphasize any serious concerns such as allergies. This will make it clear to the building management that it cannot ignore the complaint.
If management does nothing, you might consider hiring an attorney to write a letter to its central office. An attorney demand letter can have an enormous effect in generating action. Your attorney will be able to point out to the management that, in most cases, it has a legal responsibility to enforce the building rules or condo/coop agreement.